Informatic error-disturbance relation in the qubit case

TL;DR

This paper investigates the error-disturbance relation in quantum measurements from an information perspective, proposing strategies for optimal information gain and deriving bounds using information causality, with conjectures on position-momentum measurements.

Contribution

It introduces an information-theoretic approach to the error-disturbance relation, proposing cloning and swapping strategies, and derives upper bounds based on information causality.

Findings

Proposes the coarse-grained random access code model.

Derives upper bounds of information gain using information causality.

Conjectures about the information gain in position-momentum measurements.

Abstract

In 1927, Heisenberg heuristically disclosed the tradeoff between the error in the measurement and the caused disturbance on another complementary observable. In the quantum theory, most of uncertainty relations are proposed to reveal the amount of unavoidable uncertainty in the measuring process. In this paper, we study the error-disturbance relation from the information viewpoint. We ask how much information, rather than how much uncertainty, can be obtained during the two sequential measurements. To achieve optimal information gain, we argue that the strategy for the "intelligent" prior apparatus is to clone the unknown state, and for the posterior one is to perform the swapping operation. We propose the coarse-grained random access code, and therein information causality as a physical principle can be exploited for deriving the upper-bound of information gain. Finally, we conjecture the information gain of measuring the position and momentum of a quantum object in the coarse-grained way.